I think we are back to the discussion of usefulness of SE/RSE provided
by Nonmem (search in archives will recover many e-mails on the subject).
I am reluctant to disagree with Lewis Sheiner (if this indeed was his
citation that was not taken out of context) but SEs are very useful (not
perfect but useful) as an indicator of identifiability of the problem
and some measure of precision. One cannot do bootstrap on each and every
step of the modeling (moreover, bootstrap CI are not that different from
asymptotic CI, especially for well-estimated parameters). Profiling is
also local, and unlikely to give something significantly different from
the Nonmem SE results unless RSEs are very large (and profiling is as
time-consuming as bootstrap). So SEs serve as a very useful indicator
whether data support the model.
Note that Nonmem SEs are local; they rely on approximation of the OF
surface, essentially, give the curvature of this surface, so they do not
take into account any constrains. In this sense, form of the model
(CL=THETA(1) or CL=log(THETA(2)) just define the rule of extrapolation
beyond the locality of the estimate. For practical purposes, SEs of
these two representations are related (approximately) as
SE(THETA2)=SE(THETA(1))/THETA(1)
(propagation of error rule). Given SE(THETA(1)) and THETA(1) you can
estimate SE(THETA(2)) and simulate accordingly.
Thanks
Leonid
--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566
On 2/12/2015 9:04 AM, Bob Leary wrote:
Dear Aziz -
The approximate likelihood methods in NONMEM such as FO, FOCE,and LAPLACE
optimize an objective function than is parameterized internally
by the Cholesky factor L of Omega, regardless of whether the matrix is diagonal
(the EM -based methods do something considerably different and work directly
with Omega rather than
the Cholesky factor.)
Thus for the approximate likelihood methods, the SE's computed internally by
$COV from the Hessian or Sandwich or Fisher score methods
are first computed with respect to these Cholesky parameters , and then the
corresponding SE's of the full Omega=LL' are computed by a 'propagation of
errors' approach
which skews the results, particularly if the SE's are large. Thus in a sense
regarding your dilemma about whether Model 1 or Model 2 is better with respect
to applicability of $COV results, one answer is that both are fundamentally
distorted by the propagation of errors method with respect to the Omega
elements.
But regarding your fundamental question 'can we trust the output of $COV '-
all of this makes very little difference. Standard errors computed by $COV are
inherently dubious - the applicability of the usual asymptotic arguments is
very questionable for the types/sizes of data sets we often deal with.
As Lewis Sheiner used to say of these results, 'they are not worth the
electrons used to compute them'. They are the best we can do for the level
of computational investment put into them -
If you want something better, try a bootstrap or profiling method.
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